Generalized von Smoluchowski Model of Reaction Rates, with Reacting Particles and a Mobile Trap

Journal of Statistical Physics, Vol. 95, Nos. 1�2, 1999

Generalized von Smoluchowski Model of ReactionRates, with Reacting Particles and a Mobile Trap

Aleksandar Donev,1 Jeff Rockwell,1 and Daniel ben-Avraham1, 2

Received October 27, 1998

We study diffusion-limited coalescence, A+A # A, in one dimension, in thepresence of a diffusing trap. The system may be regarded as a generalization ofvon Smoluchowski's model for reaction rates, in that (a) it includes reactionsbetween the particles surrounding the trap, and (b) the trap is mobile��twoconsiderations which render the model more physically relevant. As seen fromthe trap's frame of reference, the motion of the particles is highly correlated,because of the motion of the trap. An exact description of the long-timeasymptotic limit is found using the IPDF method and exploiting a ``shielding''property of reversible coalescence that was discovered recently. In the casewhere the trap also acts as a sources��giving birth to particles��the shieldingproperty breaks down, but we find an ``equivalence principle'': Trapping and dif-fusion of the trap may be compensated by an appropriate rate of birth, suchthat the steady state of the system is identical with the equilibrium state in theabsence of a trap.

KEY WORDS: Smoluchowski model, diffusion-limited coagulation; trapping;nonequilibrium kinetics.

I. INTRODUCTION

Non-equilibrium kinetics of diffusion-limited reactions has been the subjectof much recent interest.(1�6) While equilibrium systems can be completelyanalyzed by means of standard thermodynamics methods, and reaction-limited processes are well described by classical rate equations, (7, 8) thereexist no such general approaches to the problem of nonequilibrium, diffu-sion-limited reactions.

97

0022-4715�99�0400-0097�16.00�0 � 1999 Plenum Publishing Corporation

1 Physics Department, and Clarkson Institute for Statistical Physics (CISP), Clarkson Univer-sity, Potsdam, New York 13699-5820.

2 e-mail: qd00�clarkson.edu.

A fundamental model for the reaction rate in diffusion-limited pro-cesses had been presented by von Smoluchowski.(9) In this model, an idealspherical trap is surrounded by a swarm of Brownian particles. The rate ofabsorption of particles into the spherical trap models the reaction rate. TheSmoluchowski model is limited in two important respects: (a) the particlesreact with the spherical trap but not with each other, and (b) the trap itselfdoes not diffuse, but remains static at the origin. Both limitations areunphysical: In real reaction processes all particles interact (and may react)with each other, and all particles are mobile.

Several attempts have been made to remove the restriction of animmobile trap.(10�12) The problem is complicated by the apparent correla-tions in the motion of the surrounding particles: With a mobile trap, themotion of the surrounding particles is highly correlated, since a step of thetrap to the left, say, in the lab frame of reference, results in an apparentstep to the right of all the surrounding particles in unison, in the trap'sframe of reference. Results are restricted to empirical formulas inspired bynumerical simulations, (10) or to a number of special cases (immobile par-ticles;(11) short times (12)). As regards reactions between the surroundingparticles, these could hardly be considered, other than numerically, becausefew models of diffusion-limited kinetics yield themselves to exact analysis.In fact, diffusion-limited coalescence (A+A � A) and annihilation(A+A � 0) in one dimension alone account for most of the known exactresults.(13�23)

Recently, we have studied reversible coalescence, A+A # A, on theline and in the presence of a static trap.(24) An exact analysis is possiblewith the method of interparticle distribution functions (IPDF).(25) We havefound a remarkable property of ``shielding'': The particle nearest to the trapeffectively shields the remaining particles from the trap. The steady state ofthe system is uniquely characterized by the distance of the nearest particleto the trap��all other particles remain distributed exactly as in the equi-librium state of the system in the absence of a trap. This shielding propertypersists even in the presence of a bias field (convection, or drift).

In this paper we consider reversible coalescence with a mobile trapwhich diffuses with a diffusion constant DT , not necessarily equal to thediffusion constant of the surrounding particles, D. The problem may still beformulated with the IPDF method, in spite of the correlations inducedby the motion of the trap. The shielding property of the nearest particleto the trap also remains in effect, and it enables us to find a completeexact description of the distribution of particles in the long-time asymptoticlimit. We find that relative to the trap the surrounding particles remainat equilibrium, but the gap to the nearest particle is proportional toD+DT .

98 Donev et al.

An interesting generalization is to the case when the backward processA � A+A is not limited to the particles alone, but the trap too may actas a source. Again, the rate of generation of particles from the trap, vT neednot coincide with v, the production rate from the reverse birth reaction. Wefind that if vT>0 the shielding property breaks down, and we are unableto derive a complete exact description of the system. However, if the birthrate is vT=v(1+DT�D), the effect of the trap is nullified: The particlesremain distributed as in equilibrium, as if the trap were not present.

The remainder of this paper is organized as follows. In Section II wereview the coalescence process and the IPDF method used for its analysis.The model with a mobile trap is considered in Section III. In Section IV,we generalize to the case where the trap may also act as a source. Weconclude with a summary and discussion, in Section V.

II. COALESCENCE AND THE IPDF METHOD

Our model(13, 14, 25) is defined on a one-dimensional lattice of latticespacing a. Each site is in one of two states: occupied by a particle A(v), orempty (b). Particles hop randomly to the nearest neighbor site to their rightor left, at rate D�a2. Thus, in the continuum limit of a � 0 the particlesundergo diffusion with a diffusion constant D. A particle may give birthto an additional particle, into a nearest neighbor site, at rate v�a (oneither side of the particle).3 If hopping or birth occurs into a site which isalready occupied, the target site remains occupied. The last rule meansthat coalescence, A+A � A, takes place immediately upon encounter ofany two particles. Thus, the system models the diffusion-limited reactionprocess

A+A # A (1)

The dynamical rules of the model are illustrated in Fig. 1.An exact treatment of the problem is possible through the method of

Empty Intervals, known also as the method of Inter-Particle DistributionFunctions (IPDF).(25) The key concept is En, m(t)��the probability thatsites n, n+1,..., m are empty at time t. The probability that site n isoccupied is

Prob(site n is occupied)#Prob( vn

)=1&En, n (2)

99Generalized von Smoluchowski Model of Reaction Rates

3 Our notation here differs from previous work: we take the birth rate to be v�a rather thanv�2a, to achieve a more aesthetic form of the final result.

File: 822J 228904 . By:XX . Date:07:06:07 . Time:08:47 LOP8M. V8.B. Page 01:01Codes: 1986 Signs: 1032 . Length: 44 pic 2 pts, 186 mm

Fig. 1. Reaction rules: (a) diffusion; (b) birth; and coalescence, (c) following diffusion, and(d) following a birth event. The broken lines in (a) and (b) indicate alternative target sites.

The event that sites n through m are empty (prob. En, m) consists of twocases: site m+1 is also empty (prob. En, m+1), or it is occupied. Thus, theprobability that sites n through m are empty but site m+1 is occupied, is

Prob( bn

} } } bm

v )=En, m&En, m+1 (3)

and likewise,

Prob( v bn

} } } bm

)=En, m&En&1, m (4)

With this in mind, one can write down a rate equation for the evolu-tion of the empty interval probabilities:

�En, m

�t=

Da2 (En, m&1&En, m)&

Da2 (En, m&En, m+1)

+Da2 (En+1, m&En, m)&

Da2 (En, m&En&1, m)

&va

[(En, m&En, m+1)+(En, m&En&1, m)] (5)

For example, the first term on the r.h.s. of Eq. (5) accounts for the increasein En, m when the particle at the right edge of b

n} } } b v

mhops to the right

and the sites n,..., m become empty; the second-term denotes the decreasein En, m when a particle at m+1 hops to the left into the empty intervaln,..., m, and so on.

100 Donev et al.

Equation (5) is valid for m>n. The special case of m=n correspondsto En, n��the probability that site n is empty. It is described by the equation

�En, n

�t=

Da2 (1&En, n)&

Da2 (En, n&En, n+1)+

Da2 (1&En, n)

&Da2 (En, n&En&1, n)&

va

[(En, n&En, n+1)+(En, n&En&1, n)]

(6)

Comparison with Eq. (5) yields the boundary condition: En, n&1=1. The factthat the [En, m] represent probabilities implies the additional condition thatEn, m�0. Finally, if the system is not empty then limn � &�; m � +� En, m=0.

In many applications, it is simpler to pass to the continuum limit. Wewrite x=na and y=ma, and replace En, m(t) with E(x, y, t). Letting a � 0,Eq. (5) becomes

��t

E=D \ �2

�x2+�2

�y2+ E&v \ ��x

&�

�y+ E (7)

with the boundary conditions,

E(x, x, t)=1 (8)

E(x, y, t)�0 (9)

limx � &�; y � +�

E(x, y, t)=0 (10)

The concentration becomes

\(x, t)=&�

�yE(x, y, t)|y=x (11)

and one can also show that the conditional joint probability for havingparticles at x and y but none in between, is

P2(x, y, t)=&�2

�x �yE (x, y, t) (12)

From P2 one obtains the ``forward'' (and also ``backward'') IPDF��theprobability that given a particle at x ( y) the next nearest particle to itsright (left) is at y (x):

pf (x, y, t)=\(x, t)&1 P2(x, y, t); pb(x, y, t)=\( y, t)&1 P2(x, y, t)

(13)


The IPDF method can also handle multiple-point correlation func-tions.(14) Let En(x1 , y1 , x2 , y2 ,..., xn , yn , t) be the joint probability that theintervals [xi , yi] (i=1, 2,..., n) are empty at time t. The intervals are non-overlapping, and ordered: x1< y1< } } } <xn< yn . Then, the n-pointcorrelation function (the probability of finding particles at x1 , x2 ,..., xn attime t) is given by

\n(x1 ,..., xn , t)=(&1)2 �n

�y1 } } } �ynEn(x1 , y1 ,..., xn , yn , t)|y1=x1 ,..., yn=xn

(14)

For reversible coalescence, the En satisfy the partial differential equation:

��t

En(x1 , y1 ,..., xn , yn , t)=D \ �2

�x21

+�2

�y21

+ } } } +�2

�x2n

+�2

�y2n+ En

&v _\ ��x1

&�

�y1++ } } } +\ ��xn

&�

�yn+& En

(15)

with the boundary conditions

limxi A yi or yi a xi

En(x1 , y1 ,..., xn , yn , t)=En&1(x1 , y1 ,..., x3 i , y3 i ,..., xn , yn , t)

(16)

and

limyi A xi+1 or xi+1 a yi

En(x1 , y1 ,..., xn , yn , t)=En&1(x1 , y1 ,..., y3 i , x3 i+1 ,..., xn , yn , t)

(17)

For convenience, we use the notation that crossed out arguments (e.g. x3 i)have been removed. The En are tied together in an hierarchical fashionthrough the boundary conditions (16) and (17): one must know En&1 inorder to compute En .

As a trivial example, consider the homogeneous steady state of revers-ible coalescence. This is in fact an equilibrium state, which satisfies detailedbalance. The particles are simply distributed completely randomly��a statewhich maximizes their entropy. One obtains

En, eq=exp[&#[( y1+x1)+ } } } +( yn&xn)]] (18)

and

\n, eq(x1 , x2 ,..., xn)=#n (19)

where ##v�D is the particle concentration at equilibrium.

102 Donev et al.

III. COALESCENCE WITH A MOBILE TRAP

We now consider the coalescence model but with a trap which diffuseswith a diffusion constant DT : In the discrete representation, the trap hopsto its right or left at rate DT �a2. A particle that hops into the trap isirreversibly captured by it. Similarly, when the trap hops onto an occupiedsite it captures the particle in that site. It is convenient to analyze thesystem in the trap's frame of reference. In this view, the trap remains staticat a site which we choose to be the origin; n=0. When the trap does notmove (in the lab frame of reference) the changes to En, m are described byEq. (5). A motion of the trap is perceived as a coherent opposite motion ofthe particles in the trap's reference frame. Thus, the changes to En, m due tothe motion of the trap are:

��t

(En, m)trap=DT

a2 [(En+1, m+1&En, m+1)+(En&1, m&1&En&1, m)

&(En, m&En&1, m)&(En, m&En, m+1)] (20)

For example, the first term on the r.h.s. denotes the possibility that siten+1 is occupied while the subsequent sites n+2, n+3,..., m+1 are empty,and the trap hops to the right: In the trap's frame of reference the particleat n+1 seems to hop to the left, thereby clearing the [n, m]-interval.Notice that it is important to make sure that site m+1 is empty, sinceotherwise site m would become occupied as the trap moves to the right.

Putting together all the terms in (5) and (20), and passing to thecontinuum limit, we obtain

��t

E=(D+DT) \ �2

�x2+�2

�y2+ E+2DT

�2

�x �yE&v \ �

�x&

��y+ E (21)

which is now valid in the infinite wedge 0<x< y. The term with the mixedderivative is special: it arises because of the correlated motion of the par-ticles in the reference frame of the trap.

The trap at n=0 could be realized by holding that site empty, at alltimes. Thus, it follows that E1, m=E0, m , which results in the boundary con-dition (in the continuum limit)

��x

E(x, y, t)|x=0=0 (22)



In addition, the boundary condition (10) becomes

limy � �

E(x, y, t)=0 (23)

while (8) and (9) still apply, without change.We now search for a solution of Eq. (21), with the boundary condi-

tions (8), (9), (22), and (23), in the long-time asymptotic limit, �E��t=0.It is simple to find eigenfunctions which obey Eq. (8), and other eigenfunc-tions which obey Eq. (22), but we were unable to devise a systematicmethod for finding the linear combinations that would satisfy both condi-tions simultaneously. Instead, we offer a solution based on the newly dis-covered property of ``shielding'' in the coalescence model.(24, 26)

In the steady state of the coalescence model with a static trap, it isfound that the particles are distributed randomly, exactly as in the equi-librium state of the homogeneous, infinite system (end of Section II). Thesystem is then fully characterized by p(z)��the density distribution functionof the distance between the trap and the nearest particle to the trap, z. Thenearest particle effectively shields the remaining particles from the trap(Fig. 2). As we show below, the same shielding effect takes place even whenthe trap is mobile.

Assuming that shielding holds, let E(x, y | z) be the conditional prob-ability that the interval [x, y] is empty, given that the nearest particle tothe trap is at z, then:

1, x< y<zE(x, y | z)={0, x<z< y (24)

e&#( y&x), z<x< y

Fig. 2. Schematic illustration of the shielding effect: The particles in the shaded area are dis-tributed randomly and independently from each other, as in equilibrium. The gap z betweenthe particles and the trap follows the probability density distribution p0(z).

104 Donev et al.

Hence,

E(x, y)=|�

0E(x, y | z) p(z) dz=|

�

yp(z) dz+e&#( y&x) |

x

0p(z) dz

=1&F( y)+e&#( y&x) F(x) (25)

where in the last equation we introduced the definition

F(z)#|z

0p(z$) dz$ (26)

If the hypothesis of shielding is correct, the particle concentration can beobtained from Eqs. (11) and (25):

\(x)= p(x)+#F(x) (27)

Substituting E(x, y) from Eq. (25) into Eq. (21), in the stationarylimit, the variables separate:

_(D+DT)�2

�x2 F(x)+v�

�xF(x)& e#x=_(D+DT)

�2

�y2 F( y)+v�

�yF( y)& e#y

(28)

and so, one is lead to the conclusion that

(D+DT)d 2

dz2 F(z)+vddz

F(z)=C$e&#z (29)

where C$ is a constant.The general solution of Eq. (29) is

F(z)=A+Be&#$z+Ce&#z (30)

where A, B, and C=C$D2�v2DT are constants, to be determined fromboundary conditions. From the definition of F, we have; F(0)=0,limz � � F(z)=1, and F(z)�0. The boundary condition due to thepresence of the trap (Eq. (22)) translates into dF�dz| z=0=0. Thus, we find

F(z)=1+D

DT

e&#z&D+DT

DT

e&#$z (31)


where #$#v�(D+DT). It then follows that

p(z)=v

DT

(e&#$z&e&#z) (32)

From p(z) one immediately obtains the average distance between the trapand the nearest particle: (2D+DT)�v, as well as the particle concentrationin the trap's frame of reference (Eq. (27)):

\(x)=#(1&e&#$x) (33)

The last result is similar to the one obtained for a static trap, only that thewidth of the depletion zone near the trap is 1�#$=(D+DT)�v, instead of1�#=D�v.

Our original goal of finding the empty interval probability has nowbeen achieved. Using (25) and (31), we get

E(x, y)=e&#( y&x)+D+DT

DT

e&#$y[1&e&(#&#$)( y&x)] (34)

This solution can be verified by direct substitution in Eq. (21) and in theboundary conditions (8), (22), and (23). The fact that we have found asolution proves that shielding indeed takes place, even with a mobile trap.On the other hand, we have merely shown that E(x, y) is consistent withthe shielding assumption. We now wish to show that the same is true forthe whole hierarchy of En 's, and hence for all n-point correlation functions.

If the particles are distributed as implied by shielding, then, followinga reasoning similar to that which led to Eq. (25), we should have

En(x1 , y1 ,..., xn , yn)=1&F( yn)+e&#( yn&xn)[F(xn)&F( yn&1)]

+ } } } +e&#[( yn&xn)+ } } } +( yi&xi)][F(x i)&F( y i&1)]

+ } } } +e&#[( yn&xn)+ } } } +( y1&x1)]F(x1) (35)

It is easy to confirm that these functions fulfill the boundary conditions(16) and (17). Equation (15) is also satisfied, provided that F satisfies thesame equation as above, Eq. (29), with the same boundary conditions.That is, the solution found above for F, combined with Eq. (35), solves theproblem of the En . Indeed, using Eqs. (27), (14), and (35), we find then-point correlation function:

\n(x1 ,..., xn)=\(x1) #n&1 (36)

exactly as we expect from a system with the shielding property.

106 Donev et al.

IV. COALESCENCE WITH A TRAP-SOURCE

We now wish to consider a further generalization of the trappingproblem. Suppose that the trap is imperfect, in the sense that it may alsoact as a source: the trap gives birth to A particles into the site next to it,at rate vT�a. When DT=D and vT=v the trap is identical to the rest of theparticles (only that the system is empty to the left of the trap). Thus, sucha trap-source might also be viewed as a special, tagged particle, charac-terized perhaps by a different diffusing constant and back reaction rate.The model constitutes a modest first step towards the understanding of themore realistic situation where the size of aggregates matters, and clustersdiffuse and give birth at different rates, determined by their accumulatedmass. A very recent application is to the evolution of bacterial colonies livingnear a patch of nutrients. Nelson et al., (27, 28) analyze such experiments witha diffusion-limited coalescence model with a source (modeling the nutrients)similar to ours.

The evolution equation for empty intervals in the trap-source model isidentical to that of a perfect trap (Eq. (21)). The birth of particles from thetrap affects only the boundary condition at x=0: it is no longer true thatthe trap may be realized by simply holding site n=0 empty. To derive theappropriate boundary condition we consider the total changes to En, m ,which are obtained by putting together Eqs. (5) and (20). The case of n=1needs to be considered separately, since we do not know what is E0, m . Thechanges to E1, m , including birth from the trap, add up to:

��t

E1, m=Da2 [(E1, m&1&E1, m)&(E1, m&E1, m+1)+(E2, m&E1, m)]

+DT

a2 [(E2, m+1&E1, m+1)&(E1, m&E1, m+1)+(E1, m&1&E1, m)]

&va

(E1, m&E1, m+1)&vT

aE1, m (37)

Comparison of this equation with that for general n, when n=1, yields thediscrete boundary condition:

\Da2+

va+ (E1, m&E0, m)+

DT

a2 (E1, m&1&E0, m&1)=vT

aE1, m (38)


Notice that when vT=0 this reduces to E0, m=E1, m , as for a perfect trap.Passing to the continuum limit, the boundary condition for the trap-sourcebecomes

(D+DT)�

�xE(x, y, t)|x=0=vT E(0, y, t) (39)

We now seek a solution to Eq. (21), in the stationary limit �E��t=0,and which satisfies the boundary conditions (8), (9), (23), and (39). Assum-ing that shielding holds we follow the same steps as in Section III and wearrive at exactly the same result; F(z)=A+Be&#$z+Ce&#z, only that nowthe boundary condition dF�dz| z=0=0 is replaced by:

(D+DT) e&#z ddz

F(z)| z=0=vT[1&F(z)] (40)

from Eq. (39). (Again, notice that when vT=0 one recovers the conditiondF�dz| z=0=0.)

From the boundary condition F(�)=1, we obtain A=1. Further-more, from F(0)=0 we get B=&(1+C). Finally, from the boundary con-dition due to the trap-source, Eq. (40), we get

_vT

v#$ e(#&#$) z&1& (1+C )=\vT

v#$&#+ C (41)

This condition cannot be satisfied for all z generically, and so one mustconclude that shielding does not take place in the system with a trap-source.On the other hand, for the special case that C=&1 and #$vT �v=#,Eq. (41) is satisfied. In this case F(z)=1&e&#z, which leads to E=e&#( y&x) and \(x)=#. That is, the particles are distributed exactly as inequilibrium, as if there were no trap! Thus, there exists a whole class ofstates which are equivalent to the equilibrium state of the infinite,homogeneous system, without a trap. The equivalent states are charac-terized by the relation

vT=v \1+DT

D + (42)

For these states the effect of the trap is nullified. A larger diffusivity of thetrap, DT , (i.e., a larger trapping efficiency) is exactly compensated by anincreasing rate of birth vT from the trap-source.

108 Donev et al.


An interesting case is when DT=D and vT=v. Then the trap is identicalto the surrounding particles. Notice, however, that this is not a ``special''state (Eq. (42) is not satisfied), and hence we conclude that at the equi-librium state the system seems inhomogeneous from the point of view of atagged particle. To shed some light on these baffling results, we first pointout that since the equilibrium state is homogeneous it is perceived withoutchange by any moving observer which does not interact with the particles,including random walkers. [Indeed, the equilibrium state E=e&#( y&x) is asteady-state solution of Eq. (21).] Imagine then an observer diffusingthrough the system with diffusion constant DT, and which does not inter-act with the particles. From the point of view of the observer he is static,and the average concentration of particles is constant and equal to #.Ignoring the half infinite line to his left, the observer could interpretcrossings of particles from right to left as ``trapping'' events, provided thathe also interprets crossings from left to right as ``birth'' events. Theapparent rate of birth (crossings from left to right) would be vT�a=n[(D+DT)�a2+v�a], where n=#a is the average number of particles atthe site occupied by the observer. Passing to the continuum limit, werecover the ``equivalence'' condition, Eq. (42). (Alternatively, one could usethe exact discrete result: #=v�(D+va), to obtain the discrete analogue ofthe equivalence condition: vT=v[1+DT�(D+va)].)

Although shielding breaks down when the trap acts also as a source,one may still look for a solution to the problem in more conventionalways. We were unable to find an analytic solution; however, the discreteequations can be integrated numerically, and the particle system may also

Fig. 3. Concentration profile with a trap-source: Shown are results from numerical simula-tions (symbols) on a lattice of 25,000 sites, averaged over a2�D=106 time steps; as well asresults obtained from numerical integration of the exact discrete equations (solid curves). Thecases shown are for vT �v=2, 1, 0.5, and 0 (top to bottom), with DT=0.


be simulated on a computer. Computer simulations confirm the fact thatshielding breaks down when vT>0, and that the particle concentrationbeyond the nearest particle to the trap is no longer as in equilibrium. InFig. 3 we show typical results for various values of vT . As vT increases, theconcentration of particles near the trap increases, from zero (for vT=0),to # (for the appropriate ``special'' rate, Eq. (42)), and to concentrationslarger than #.

V. DISCUSSION

We have studied diffusion-limited coalescence, A+A # A, in thepresence of a diffusing trap, and we have found an exact description of thelong-time asymptotic limit, using the IPDF method. When the trap is per-fect, the system displays a ``shielding'' property: the particle nearest to thetrap effectively shields the other particles from the trap. That is, the par-ticles remain distributed as in the equilibrium steady-state of the infinitehomogeneous system (without a trap), and only the distance between thetrap and the nearest particle is unusual. This distance grows linearly withDT��the diffusion coefficient of the trap.

For an imperfect trap which also acts as a source the shieldingproperty breaks down and we were unable to find an analytic solution, butthe exact equations can then be solved numerically. We have found anintriguing ``equivalence principle'': all systems with vT=v(1+DT �D) areequivalent to each other. The trap then seems invisible and the particlesremain distributed as in the homogeneous equilibrium state.

Our system is a generalization of von Smoluchowski's model for reac-tion rates; the particles react with each other, and the trap is mobile. Thereaction rate equals the rate of influx of particles into the trap. This rateis k=n[(D+DT)�a2+v�a], where n is the average number of particles atthe site adjacent to the trap. For a perfect trap, we use the result ofEq. (33) to find n=a2(d\�dx)x=0=a2##$=a2v2�D(D+DT), and so, in thecontinuum limit (a � 0) we get k=v2�D. Curiously, the trapping rate isindependent of the diffusivity of the trap, DT . A faster trap visits more sitesper unit time, but it also depletes its immediate neighborhood more effec-tively, and the two effects cancel each other. For the case of a trap-source,we have failed to obtain an analytic expression for n, and hence we couldcompute k only numerically.

There remain several interesting open problems. We have consideredonly the steady state of our model, but the transient is also of interest. Forperfect traps the shielding property holds at all times (provided that theinitial condition is compatible with it) and one can exploit it to find ananalytic answer. An important open problem is that of finding a systematic

110 Donev et al.

method for solving the evolution equation for empty intervals. We werefortunate to come across a solution which obeys shielding, but shieldingdoes not always hold, as for example for non-ideal trap-sources. Indeed,most remaining open questions concern the model with a trap-source. Anexact analytic solution for this case is still missing. We have managed toprove, however, that the solution could not be of the form of a sum ofexponentials (finite or infinite), other than for the special states, equivalentto equilibrium.

An interesting question is whether there exist other classes of equiv-alence. That is, are there any states equivalent to each other, but not to theequilibrium state?��we have managed to prove that such states do notexist, at the level of the empty interval probability. However, there remainsthe possibility that different systems might share the same concentrationprofile, in spite of differences in their empty interval probabilities. Whethersuch states exist remains an open challenge.

ACKNOWLEDGMENTS

We would like to thank NSF for support of this work.

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Generalized von Smoluchowski Model of Reaction Rates, with Reacting Particles and a Mobile Trap

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